Stability of a diamond-shaped module with multiple interaction types: Electronic Supplementary Material (ESM)
S1: Indirect effects between resource species when interactions are asymmetric
As mentioned in the paper, the nature of the indirect effects occurring between the two resource species depends on the symmetry of the interactions. If the relative effects that two species have on each other are the same, then the indirect effects have the same sign and magnitude (Cij = Cji). Such conditions result either on apparent facilitation or on apparent competition. If the interactions are asymmetric, then the indirect effects can differ in sign and/or magnitude. Such conditions can result on apparent antagonism (Cij>0 and Cji<0).
To ease our understanding, we do not consider all interactions to be asymmetric simultaneously and detail different possible cases. In the following cases, we suppose all interactions to have the same potential strength I, and.
Case 1: P1 has asymmetric interactions with M and H, P2 has symmetric interactions with M and H
Let εM be the relative strength of the effect of M on P1, and εH the relative strength of the effect of H on P1. Conversely, (1-εM) is the relative strength of the effect of P1 on M, and (1-εH) is the relative strength of the effect of P1 on H. Then,, , ,. Moreover, we assume that interactions involving P2 all have the same strength:. The indirect effects that P2 has on P1 is then. Consequently, there is apparent antagonism if εM≠εH. This condition means that the effect of P1 on the mutualist is not the same on the exploiter. If εMεH then P2 has a positive indirect effect on P1 whereas P1 has a negative effect on P2. If εMεH, then P2 has a negative indirect effect on P1 whereas P1 has a positive indirect effect on P2.
Case 2: The antagonistic interactions are asymmetric whereas the mutualistic interactions are symmetric
We name here εP1 and εP2 the relative magnitude of the effect of the exploiter respectively on P1 and P2. Conversely, 1-εP1 and 1-εP2 are the relative magnitudes of the effect of P1 and P2 on the exploiter. Then,,,, and. Moreover, we assume that all mutualistic interactions have the same strength:. The indirect effects that Pj has on Pi is then. If, then Cij>0 and Cji>0: there is apparent competition. In this case, the effects of H on P1 and P2 are close (Fig. 1). If and or and, then Cij>0 and Cji<0: there is apparent antagonism. In this case, the effects of H on P1 and P2 are rather different. The antagonistic interactions are asymmetric, and furthermore, either H has a strong effect on P2 whereas it has a weak effect on P1, or the opposite.
S2: Effect of the generalism of the mutualist and the antagonist on community persistence and stability
In this section, we consider that all interactions are symmetric and of similar magnitude. In such case, the conditions for a stable coexistence of all species depend on the generalism of the mutualist and the antagonist. By following the same assumptions as in section 3.1, we detail conditions (i), (ii) and (iii):
Condition (i): Net intra-specific competition for resource species. If we consider first the impact of generalism, we can express Cii and Cjj as a function of the relative preference of the consumer and the mutualist for species i: and where corresponds to the difference between the preferences of H and M for the resource species i. In the cases of apparent facilitation and apparent competition, condition (i) requires Cii>0 i{1,2}. Condition (i) is always satisfied for both resource species if both the consumer and the mutualist prefer equally the same resource species. To the contrary, differences in resource preferences of H and M can lead to a negative intra-specific net effect for one resource species.
Condition (ii): Combined intra-specific net effects greater than combined net inter-specific effects. When the effects of interactions with the mutualist and the consumer are symmetric and of similar magnitude, condition (ii) depends strongly on the differences in resource preferences of H and M: where . In such case, condition (ii) is likely to be met if Di is small, that is if preferences of the mutualist and the consumer for resource species i are not too different. If Di is not null, condition (ii) is also more likely to be met if is positive. This can be realised either if (and ) or if (and ). These two cases correspond respectively to the left and right quadrants of Fig. 2 when the mutualist is more generalist than the consumer, i.e. when there is apparent facilitation between the resource species.
Condition (iii): Condition for species coexistence. When interactions are symmetric and of similar magnitude, condition (iii) needs and where and. In such case, as for condition (ii), condition (iii) is more likely to be met if Di is small, that is if preferences of the mutualist and the consumer for resource species i are not too different. If Di is not null, condition (iii) is also more likely to be met if and are positive. As for condition (ii), this requires the mutualist to be more generalist than the consumer, leading to apparent facilitation between the resource species.
S3: Effect of interaction asymmetry on community persistence and stability
In this section, we consider that mutualistic and antagonistic interactions are of similar magnitude. In such case, stable coexistence of all species depends on the level of asymmetry of interactions. By following the same assumptions as in section 3.2, we detail conditions (i), (ii) and (iii):
Condition (i): Net intra-specific competition for resource species. When mutualistic interactions are asymmetric but not antagonistic interactions, intra-specific net effects can be written as follows: and . AsεPi[0,1] by its definition, is always positive. Then, intra-specific net effects are always positive in this case. To the contrary, if antagonistic interactions are asymmetric but not mutualistic interactions, . Intra-specific net effects can be negative if the interactions are too asymmetric. Condition (i) is then satisfied for both resource species as soon as the interactions between the resource species and the consumer are not too asymmetric. Please note that when interactions are not too asymmetric, and then the effects of mutualistic and antagonistic interactions on a given resource are close, (i∈{1,2}). Hence, this is a particular example of a balance between mutualistic and antagonistic effects on resource species.
Condition (ii): Combined intra-specific net effects greater than combined net inter-specific effects. When only mutualistic interactions are asymmetric, condition (ii) can be written as follows. Condition (ii) is always satisfied in this case since we always have. If only antagonistic interactions are asymmetric, condition (ii) is written as. Condition (ii) is more likely to be satisfied when the asymmetry of interaction is not too high and if interactions of the two resource species with the consumer differ in their levels of asymmetry.
Condition (iii): Condition for species coexistence. When mutualistic interactions are asymmetric but not antagonistic interactions, condition (iii) can be written as follows (with the same assumptions as in section 3.2): and. In such case, condition (iii) is more likely to be satisfied when the asymmetry of the two mutualistic interactions do not differ strongly and when the mutualistic interactions favour more the resource species than the mutualist (). When only antagonistic interactions are asymmetric, condition (iii) is almost the same as for the latter case: and. It is more likely to be satisfied when the asymmetry of the two antagonistic interactions do not differ strongly and when the antagonistic interactions affects less the resource species than it benefits the antagonist ().
S4: Numerical application of stability in a community with asymmetric interactions
Under the hypotheses of asymmetric interactions either between the mutualist and the resource species, or between the consumer and the resource species, we measure the stability of the community for different distributions of relative effects of the mutualist and the consumer on resource species.
Fig. S4.1: Effect of the asymmetry of mutualistic interactions on community stability. Stability of the community module for different distributions of the interaction strengths, here expressed as the relative effects of the mutualist on resource species. We thus consider how the asymmetry of mutualistic interactions affect community stability. Stability is measured by the highest real part of the eigenvalue of the Jacobian matrix of the whole community, i.e. the Jacobian matrix whose elements are the effects that each of the four species has on each other. We thus release the assumption of a fast-slow system whose dynamics are driven by resource species. The darker is the colour, the more stable is the community. The redder is the colour, the less stable is the community. White zones show areas where the conditions for stability and feasibility are not fulfilled. The white lines delimit conditions for apparent antagonism. a) Model parameters are = 1; = 10; =5; =50; I = 10, b) = 5; = 10; =5; =50; I = 1, c) = 5; = 10; =5; =50; I = 10.
Fig. S4.2: Effect of the asymmetry of antagonistic interactions on community stability. Stability of the community module for different distributions of the interaction strengths, here expressed as the relative effects of the exploiter on resource species. We thus consider how the asymmetry of antagonistic interactions affect community stability. Stability is measured by the highest real part of the eigenvalue of the Jacobian matrix of the whole community. We thus release here the assumption of a fast-slow system whose dynamics are driven by resource species. The darker is the colour, the more stable is the community. The redder is the colour, the less stable is the community. White zones show conditions when stability and feasibility are not fulfilled. As in fig. A2, the white lines delimit the conditions for apparent antagonism. a) Model parameters are = 1; = 10; =5; =50; I = 10, b) = 5; = 10; =5; =50; I = 1, c) = 5; = 10; =5; =50; I = 10.
When mutualistic interactions are not symmetric, stability is higher when the asymmetry is at the advantage of the resource species. Such condition correspond to apparent competition between the two resource species (Fig. S4.1). When antagonistic interactions are not symmetric, stability is higher when asymmetry is high and the effect of the consumer on the resource species being strong. Such conditions correspond to apparent facilitation between the two resource species (Fig. S4.2). Although asymmetry in such module can be much more complex, these results suggest that apparent antagonism could promote less stability than apparent competition and apparent facilitation.
S5: Analysis of feasibility and stability of separated apparent facilitation and apparent competition modules
Hereafter, we compare the conditions for feasibility and stability of the separated apparent facilitation and apparent competition modules. For a better understanding of our main results, we make the same assumptions on the dynamics of the separated modules as for the diamond-shaped module analysed in the main text. We thus model the dynamics of each separated module as a fast-slow dynamical system, assuming that the mutualist M or the consumer H have faster dynamics than the resource species.
Apparent facilitation module:
Once the mutualist has achieved its equilibrium state, we can write the resource dynamics as follows:
where the net productivity of resource species i, the net effect of resource species j on resource species i, and the net effect of resource species i on itself. The fixed point is:
Given , it is feasible if, or if,.
Let J* be the Jacobian matrix at the equilibrium of the fast-slow version of this module:
The community is stable ifand. Thus, the system is feasible and stable if Cii>0 (1), (2), and (3).
Apparent competition module:
Once the exploiter has achieved its equilibrium state, we can write the resource dynamics as follows:
where the net productivity of resource species i, the net effect of resource species j on resource species i, and the net effect of resource species i on itself. The fixed point is:
Given , it is feasible if,or if ,.
Let J* be the Jacobian matrix at the equilibrium of the fast-slow version of this module:
The community is stable ifand. Thus, the system is feasible and stable if Cii>0 (4), (5), and (6). Conditions (4) and (5) are always true in such module.
We compare the two types of modules.
Note that among the feasibility conditions in these modules, we need and. If we suppose all the parameters of these model to be the same, in other words, the ecological situations to be the same except the type of interaction, the apparent facilitation module is more likely to be feasible than the apparent competition module. However, given that (4) and (5) are always true, the apparent competition module is more likely to be stable than the apparent facilitation module as soon as feasibility conditions are satisfied.